Universal code of the positive integers based on factorization

摘要

It is known that any universal code of the positive integers must satisfy L(n) ≥ log{sub}2{sup}* n - (log{sub}2{sup}2 e)ω{sup}*(n) for infinite many positive integers n, where L(n) stands for the codeword length, ω{sup}*(n) is the largest integer s such that log{sub}2{sup}s n≥0, and log{sub}2{sup}* n=log{sub}2 n+log{sub}2{sup}2 n+... + log{sub}2{sup}(ω{sup}*(n))(n). On the other hand, any known universal codes satisfy this inequality of L(n) for all large integers. But this fact does not deny that some code has L(n) shorter than the lower bound for infinite many integers. In this paper, for any given code C(n), we construct a new code C{top}～(n) based on factorization of n, and we show that the codeword length of C{top}～(n), L{top}～(n), satisfies that L{top}～(n) ≤ L(n) +2L(1) for all n, and L{top}～(n)